Results 1 to 6 of 6

Math Help - homeomorphism help

  1. #1
    Junior Member
    Joined
    Dec 2008
    Posts
    70

    homeomorphism help

    Hi, learning about homemorphic functions at the moment and am failry confused,

    If i have a homeomorphic function h: A - B between two topological spaces A and B,


    Is the pre image of the empty set in B the empty set in A ?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,391
    Thanks
    758

    Re: homeomorphism help

    let me ask you this: can the pre-image of the empty set in B have ANY elements in it?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Dec 2008
    Posts
    70

    Re: homeomorphism help

    Is it that it cant because a homeomorphic function is bijective and continuous?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member Sylvia104's Avatar
    Joined
    Mar 2012
    From
    London, UK
    Posts
    107
    Thanks
    37

    Re: Homeomorphism help

    Look closely at the definition of pre-image: If f:A\to B and V\subseteq B, the pre-image of V under f is f^{-1}(V)=\{x\in A:f(x)\in V\}. In other words, x\in f^{-1}(V) \Leftrightarrow f(x)\in V for all x\in A. Now try assuming f^{-1}(\O)\ne\O and see what happens.
    Last edited by Sylvia104; May 16th 2012 at 03:11 AM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Dec 2008
    Posts
    70

    Re: Homeomorphism help

    i'm sorry i understand the definition but i think i may have some understanding of the empty set missing, obviously this cant happen but why cant a function map something to the empty set and the empty set to something else
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member Sylvia104's Avatar
    Joined
    Mar 2012
    From
    London, UK
    Posts
    107
    Thanks
    37

    Re: Homeomorphism help

    Quote Originally Posted by monster View Post
    why cant a function map something to the empty set and the empty set to something else
    We're not talking about the empty set as an element of A or B; we're talking about it as a subset of A and B.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Homeomorphism
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: June 11th 2011, 11:00 AM
  2. Homeomorphism
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: March 23rd 2011, 11:49 AM
  3. homeomorphism between R^(n^2) and M_n(R)
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: January 21st 2011, 05:20 PM
  4. homeomorphism
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: November 30th 2009, 01:29 PM
  5. Homeomorphism
    Posted in the Calculus Forum
    Replies: 7
    Last Post: March 3rd 2008, 01:57 AM

Search Tags


/mathhelpforum @mathhelpforum